Vertex Equitable Labeling of Signed Bistars

Abstract

A signed graph\(S=(S^u,\sigma )\) has a graph \(S^u\) and a function \(\sigma :E(S^u)\longrightarrow \{+,-\}\).
Let S has q edges and \({\mathcal {A}} = \{0,1,2,\ldots , \lceil \frac{q}{2}\rceil \}\). A vertex labeling \(f:V(S)\longrightarrow {\mathcal {A}}\) which is onto, is said to be a vertex equitable labeling of S if it induces a bijective edge labeling \(f^*:E(S)\longrightarrow \{1,2,\ldots ,{\mathfrak {m}},-1,-2,\ldots ,-{\mathfrak {n}}\}\) defined by \(f^*(uv) = \sigma (uv)(f(u)+f(v))\) such that \(|v_f(a)-v_f(b)|\le 1\), \(\forall a, b\in {\mathcal {A}}\), where \(v_f(a)\) is the number of vertices with \(f(v) = a\) and \({\mathfrak {m}}, {\mathfrak {n}}\) are number of positive and negative edges respectively in S. A signed graph S is said to be vertex equitable if it admits a vertex equitable labeling. In this paper, we study vertex equitable behavior of signed bistars.